Characterizations for S-Convex-Averaging Domains via Two-Dimensional Diffusion-Wave Equations

In this paper, we introduce the concept of s-convex-averaging domains, which are extensions of circular and irregular convex domains, by using s-convex functions and generalized Orlicz norms. Based on the quasi-hyperbolic metric and Lp-averaging domains, several fundamental properties of s-convex-averaging domains are characterized. These properties are applied to the domains of a class of two-dimensional diffusion-wave equations. Furthermore, we establish intrinsic relationships between the considered partial differential equations and the geometric structure of s-convex-averaging domains. Finally, the embedding inequality for the solutions of these kinds of partial differential equations is derived.